One might wonder why, when moving from the standard world to the
non-standard world,
elements of sets become "*-elements" or "pseudo-elements", but
equality remains equality. This is because equality is not a
predicate! Rather, it serves to identify objects precisely (for instance, any
model guarantees that the
formula "
x=A" (where A is some constant and
x a
variable) is
true for precisely one
x). As such, it has a specific interpretation in the model.
This technical issue has various consequences. But since anything provable in the non-standard world is also provable in the standard one, clearly there's no particular cause for concern.